Process and Reality — the Gifford Lectures, 1929

(viii) If A and B be members of the ovate class, and A include B, then there are members of the class which include B and are included in A.

(ix) There are dissections (cf. Def. 4 of the previous chapter) of every member of the ovate class, which consist wholly of members of that class; and there are dissections consisting wholly or partly of members not belonging to that class.

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(b) The Abstractive Group

(i) Among the members of any point, there are ovate abstractive sets.

(ii) If any set of two, or of three, or of four, points be considered, there are abstractive sets ‘prime’ in reference to the twofold condition, (a) of covering the points in question, and (b) of being equivalent to an ovate abstractive set.

• (iii) These are sets of five points such that no abstractive set exists prime in reference to the twofold condition,
• (a) of covering the points in question, and (b) of being equivalent to an ovate abstractive set.

By reason of the definitions of this latter group, the extensive continuum in question is called ‘four-dimensional.’ Analogously, an extensive continuum of any number of dimensions can be defined. The physical extensive continuum with which we are concerned in this cosmic epoch is four-dimensional. Notice that the property of being ‘dimensional’ is relative to a particular ovate class in the extensive continuum. There may be ‘ovate’ classes satisfying all the conditions with the exception of the ‘dimensional’ conditions. Also a continuum may have one number of dimensions relating to one ovate class, and another number of dimensions relatively to another ovate class.

Possibly physical laws, of the type presupposing continuity, depend on the interwoven properties of two, or more, distinct ovate classes.

SECTION III

Assumption 1. In the extensive continuum of the present epoch there is at least one ovate class, with the characteristics of the two groups, (a) and (b), of the previous section.

Defintion 2. One such ovate class will be denoted by a: all definitions will be made relatively to this selected ovate class.

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It is indifferent to the argument whether or no there be an alternative ovate class. If there be, the derivative entities defined in reference to this alternative class are entirely different to those defined in reference to q. It is sufficient for us, that one such class interests us by the importance of its physical relations.

Assumption 2. If two abstractive sets are prime in reference to the same twofold condition, (a) of covering a given group of points, and (b) of being equivalent to some ovate abstractive set, then they are equivalent.

By reason of the importance of this proposition a proof is given.

Proof. The two abstractive sets are either equivalent to the same ovate abstractive set, or to different ovate abstractive sets. In the former alternative, the required conclusion is obvious. In the latter alternative, let and v be the two different ovate abstractive sets. Each of these sets, p and y, satisfies the twofold condition. We have to prove that they are equivalent to each other. Let M and N be any regions belonging to p and y respectively. Then since the convergent portions of the abstractive sets belonging to the various points of the given group must ultimately consist of regions all lying in M and all lying in N, it follows that M and N intersect. But, being oval, M and N have only one intersect, and all the points in question must be situated in it. Also this intersect is oval. Hence, by selecting such intersects, a third abstractive set can be found which satisfies the twofold condition and is covered both by y and by y. But since y and vy are prime in reference to this condition, they are both of them equivalent to this third abstractive class. Hence they are equivalent to each other. Q.E.D.

Corollary. It follows that all abstractive sets, prime with respect to the same twofold condition of this type, belong to one geometrical element.

Definition 3. The single geometrical element defined, as in the enunciation of Assumption 2, by a set of two points is called a ‘straight’ segment between those endp. 466points. If the set comprise more than two points, the geometrical element is called ‘flat.’ ‘Straight’ segments are also included under the designation ‘flat geometrical elements.’

If a set of points define a flat geometrical element, as in the enunciation of Assumption 2, it may happen that the same geometrical element is defined by some sub-set of those points. Hence we have the following definition:

Definition 4. A set of points, defining a flat geometrical element, is said to be in its lowest terms when it contains no sub-set defining the same flat geometrical element.

Assumption 3. No two sets of a finite number of points, both in their lowest terms, define the same flat geometrical element.

Definition & The locus of points incident in a ‘straight segment’ is called the ‘straight line’ between the endpoints of the segment.

Definition 6. The locus of points incident in a flat geometrical element is called the ‘content’ of that element. It is also called a ‘flat locus.’

Assumption 4. If any sub-set of points lie in a flat locus, that sub-set also defines a flat locus contained within the given locus.

Definition 6. A complete straight line is a locus of points such that, (i) the straight line joining any two members of the locus lies wholly within the locus, (ii) every sub-set in the locus, which is in its lowest terms, consists of a pair of points, (ili) no points can be added to the locus without loss of one, or both, of the characteristics (i) and (ii).

Definition 7. A triangle is the flat locus defined by three points which are not collinear. The three points are the angular points of the triangle.

Definition 8. A plane is a locus of non-collinear points such that, (1) the triangle defined by any three non-collinear members of the locus lies wholly within the locus,

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(ii) any finite number of points in the locus lie in some triangle wholly contained in the locus, (ili) no set of points can be added to the locus without loss of one, or both, of the characteristics (i) and (ii).

Definition 9. A tetrahedron is the flat locus defined by four points which are not coplanar. The four points are called the corners of the tetrahedron.

Definition 10. A three-dimensional flat space is a locus of non-coplanar points such that, (i) the tetrahedron defined by any four non-coplanar points of the locus lies wholly within the locus, (ii) any finite number of points in the locus lie in some tetrahedron wholly contained in the locus, (i11) no set of points can be added to the locus without the loss of one, or both, of the characteristics (i) and (ii).

Any further development of definitions and propositions will lead to mathematical details irrelevant to our immediate purposes. It suffices to have proved that characteristic properties of straight lines, planes, and three-dimensional flat spaces are discoverable in the extensive continuum without any recourse to measurement. The systematic character of a continuum depends on its possession of one or more ovate classes. Here, the particular case of a ‘dimensional’ ovate class has been considered.

SECTION IV

The importance of the notion of ‘external connection’ requires further discussion.

First, there is a purely geometrical question to be noted. The theory of the external connection of oval regions throws light on the Euclidean concept of ‘evenness.’ A pair of ovals (cf. Sect. III) can only be externally connected in a ‘complete locus,’ or in a single point. We now consider that species of ‘complete loci’ which can be the points common to the surfaces of a pair of ovals externally connected. We exclude the case of onepoint contact. The species seems to have what the

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